You can play Xbox 360 games on your PC with an emulator called Xenia. Xenia is a Windows program that replicates the Xbox 360 console in software form so that you can play Xbox 360 games on your PC.

These are the button mappings for an Xbox 360 controller. The emulator is a JavaScript Game Boy emulator. This is loaded on the Chromecast through a simple html file (called the receiver in the repo).